The Gamma process and its generalizations for describing age- and/or state-dependent degradation phenomena

For several highly reliable products, it is often difficult to evaluate their reliability using classical failure time-based methods due to the low number of observed failures. Fortunately, many such products have performance characteristics whose degradation over operating time can be related to their reliability, so that, if the degradation phenomenon is adequately modelled, it is possible to accurately evaluate the reliability of these products. In addition, an accurate modelization of the degradation phenomenon allows one to predict the remaining life and the residual reliability, and then allows one to plan a condition-based maintenance policy, which can be more effective both than an age-based preventive maintenance and then a corrective maintenance policy. A widely used model to describe degradation phenomena is the non-stationary Gamma process, which however is not always suitable to describe the observed processes. Of course, a wrong modeling of the degradation process leads to an inaccurate estimate of the product reliability and a wrong determination of the optimal maintenance policy. Thus, this talk aims to recognize some common situations where the Gamma process is not suitable and to suggest some alternative models, such as the time-discrete Extended Gamma process, the Inverse Gamma process, and the Transformed Gamma process, that can be viewed as generalizations of the Gamma process and can find application where the Gamma process is not suitable. Numerical applications of such processes to real data sets are also briefly discussed.